Pronormal subgroup

A subgroup $H$ of a group $G$ satisfying the following condition: If $K$ is a subgroup in $G$ conjugate with $H$, then $K$ is conjugate with $H$ in the subgroup generated by $H$ and $K$ (cf. Conjugate elements). Sylow subgroups in finite groups, as well as Hall and Carter subgroups in finite solvable groups, are pronormal (cf. Sylow subgroup; Hall subgroup; Carter subgroup). The concept of a pronormal subgroup is closely connected with that of an abnormal subgroup. Every abnormal subgroup is pronormal, and the normalizer of a pronormal subgroup (cf. Normalizer of a subset) is abnormal.

References

[1]	L.A. Shemetkov, "Formations of finite groups" , Moscow (1978) (In Russian)

References

[a1]	D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)